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A couple simple linear algebra questions.

  1. Dec 13, 2012 #1
    I think I know the answer to these questions, but I jsut want to make sure.

    1) If A is invertible then A+A is invertible. True/False
    True.

    Because det(A)≠0, det(A+A)=det(2A)=2^n * det(A)≠0

    Is this correct.

    2) A 3x3 matrix can have 2 distinct eigenvalues. True/False
    True, although I was kind of confused with what "distinct" means.
    The characteristic polynomial can look something like this: (λ-1)^2 * (λ+2)
    Distinct just refers to the # of "different" eigenvalues right? And doesn't include them again if they're repeated?
     
  2. jcsd
  3. Dec 13, 2012 #2

    Zondrina

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    Yup, you first one looks good. As does your second one.

    Distinct does indeed refer to the amount of DIFFERENT eigenvalues a matrix has in this case.

    In general, if you have an nxn matrix, it CAN have as many as n distinct eigenvalues.
     
  4. Dec 13, 2012 #3
    so basically 0≤λ≤n?
     
  5. Dec 13, 2012 #4

    Zondrina

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    What does this statement mean? What is λ?

    I think what you're intending is to say that there are going to be greater than or equal to zero eigenvalues for a nxn matrix, this is incorrect to assume. Is a matrix really a matrix if n = 0? Even the statement makes me cringe.

    Really, we're only concerned with the eigenvalues of matrices for n ≥ 2 and only square matrices are considered.

    So supposing that A is an nxn matrix. The fewest eigenvalues it can have is 1 and that's only if the algebraic multiplicity of the eigenvalue equals n. Otherwise it can have up to n eigenvalues. I hope that clears that up for you :).
     
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